Approximate Critical Values Research Articles

Goodness-of-Fit Test for Exponentiality Based on Kullback–Leibler Information

In this paper, we discuss goodness-of-fit tests of the exponential distribution based on Kullback–Leibler information. To construct test statistics, Van Es' and Correa's entropy estimator are used as an estimator of Shannon's entropy. Critical values for various sample sizes determined by means of Monte Carlo simulations are presented for each of test statistics. Also, formulas for calculating approximate critical values for given sample sizes are provided for the respective test statistics. A Monte Carlo power analysis is performed for various alternatives and sample sizes in order to compare the proposed tests with several existing goodness-of-fit tests based on the empirical distribution function (EDF). Simulation results show that Kullback–Leibler information tests have higher powers than the EDF tests.

Detecting an innovative outlier in a set of time series

Tests for an innovative outlier affecting every member of a set of autoregressive time series at the same time point are developed. In one model, the outliers are represented as independent random effects; likelihood ratio tests are derived for this case and simulated critical values are tabulated. In a second model, assuming that the size of the outlier is the same in each series, a standard regression framework can be used and correlations between the series are introduced. Simulation studies show that approximate critical values obtained from the χ 1 2 distribution work well for heteroscedastic independent series and for the case of equal correlations between each pair of series.

Significance tables for the exact variance test for the Poisson distribution with alternative underdispersion

Fisher (1950) introduced the variance or dispersion index test statistic to test deviations of the Poisson distribution. For this test approximate critical values exist for large sample sizes. If the number of observations is small this approximation can lead to a wrong conclusion. For small samples, the exact critical values can only be derived by enumeration of all possibilities. Tables of critical values for overdispersion already exist (e.g., Rao and Chakravarti, 1956) However, in many biological situations underdispersion, a more-regular-than-Poisson distribution, is a common phenomenon. Therefore, we have tabulated in this paper the one-tailed critical values for a small number of observations under the null hypothesis (H0) that the random variable is Poisson distributed against the alternative hypothesis of underdispersion. With the help of this table, the hypothesis that the observations in a data set are Poisson distributed, can be tested easily with the variance test. The tables are illustrated with examples from the literature and some observations from our own research. In general, the χ2 approximation gives a smaller significance level than the exact variance test.

On detecting change in likelihood ratio ordering

This article studies the problem of testing and locating changepoints in likelihood ratios of two multinomial probability vectors. We propose a binary search procedure to detect the changepoints in the sequence of the ratios of probabilities and obtain the maximum likelihood estimators of two multinomial probability vectors under the assumption that the probability ratio sequence has a changepoint. We also give a strongly consistent estimator for the changepoint location. An information theoretic approach is used to test the equality of two discrete probability distributions against the alternative that their ratios have a changepoint. Approximate critical values of the test statistics are provided by simulation for several choices of model parameters. Finally, we examine a real life data set pertaining to average daily insulin dose from the Boston Collaborative Drug Surveillance Program and locate the changepoints in the probability ratios.

The distribution of test statistics for outlier detection in heavy-tailed samples

We investigate the asymptotic behavior of outliers test samples statistics for drawn from heavy-tailed distributions. We extend classical results of David et al. [1] and Grubbs [2], who considered outlier test statistics for the finite-variance case, to the heavy-tailed infinite variance case. Our main result concerns the limiting distribution of n − 1 2 O n for the outlier statistic ▪ when the observations X i are the domain of attraction of an α-stable law. We present approximate critical values for O n for finite samples using response surface methods.

An Energy Curve Family and the Transition Point of the Simple Cubic Layer Potts Model

It was shown in our previous numerical study 1) for the semi-infinite square lattice (l × ∞) of the q-state Potts model 2), 3) with periodic boundary conditions that an energy E (K, l)/J = −KdlnΛ0 (K, l)/dK versus coupling parameter K = J/kT curve family has a common intersection point, where Λ0 (K, l) is the largest eigenvalue of the transfer matrix. 4) 6) The values of E (K, l) and K = J/kT at the intersection even for small l exactly (to fourteen decimal places) give the critical ones of the bulk lattice (∞ × ∞), which are independent of l, without reference to the duality argument. 2), 3), 7) The energy curve intersection procedure (ECIP) was similarly applied to the semi-infinite simple cubic lattice (l × m × ∞) of the q-state Potts model for obtaining the approximate critical values of the layer lattice (l ×∞×∞). 8) The result of improved precision computations for the simple cubic layer lattice (l×∞×∞) is reported in the present paper. An energy curve family of E (K, l, m) versus K for the semi-infinite lattice (l×m×∞) has seemingly a common intersection point when the calculation is done up to the three decimal places. Higher precision computations show, however, that the intersection point slightly and gradually moves depending on the size m (keeping l fixed) and seems to approach the exact critical transition point of the layer lattice (l×∞×∞). Let Km be the coupling parameter at the intersection of the energy curves E (K, l, m) and E (K, l, m+ 1). As an example, Fig. 1 shows the relation between the parameter Km and the size m for the q = 3 Potts lattice (l × m × ∞) with l = 2 obtained from the intersection points of the energy curves of m and m+ 1 (4 ≤ m ≤ 7). Extrapolating from such figures of Km, we may estimate the critical values (Kc)l of the layer l lattice. For example, for the q = 2 and 3 lattices with l = 2, we have (Kc)l=2 = 0.5197 and 0.6037, respectively. The specific heat exponent μ of the simple cubic Potts lattice with l layers is evaluated by the ECIP with improved precision. The per site specific heat (Cv/k)m is estimated from the slope at the intersection of the energy curves of m and m+ 1. Making use of the curves of (Cv/k)m versus m, we find that the exponent μ of the

Critical Values for Cointegration Tests in Heterogeneous Panels with Multiple Regressors

In this paper we describe a method for testing the null of no cointegration in dynamic panels with multiple regressors and compute approximate critical values for these tests. Methods for non-stationary panels, including panel unit root and panel cointegration tests, have been gaining increased acceptance in recent empirical research. To date, however, tests for the null of no cointegration in heterogeneous panels based on Pedroni (1995, 1997a) have been limited to simple bivariate examples, in large part due to the lack of critical values available for more complex multivariate regressions. The purpose of this paper is to fill this gap by describing a method to implement tests for the null of no cointegration for the case with multiple regressors and to provide appropriate critical values for these cases. The tests allow for considerable heterogeneity among individual members of the panel, including heterogeneity in both the long-run cointegrating vectors as well as heterogeneity in the dynamics associated with short-run deviations from these cointegrating vectors.

Critical Values for Cointegration Tests in Heterogeneous Panels with Multiple Regressors

I. INTRODUCTION In this paper we describe a method for testing the null of no cointegration in dynamic panels with multiple regressors and compute approximate critical values for these tests. Methods for non-stationary panels, including panel unit root and panel cointegration tests, have been gaining increased acceptance in recent empirical research. To date, however, tests for the null of no cointegration in heterogeneous panels based on Pedroni (1995, 1997a) have been limited to simple bivariate examples, in large part due to the lack of critical values available for more complex multivariate regressions. The purpose of this paper is to fill this gap by describing a method to implement tests for the null of no cointegration for the case with multiple regressors and to provide appropriate critical values for these cases. The tests allow for considerable heterogeneity among individual members of the panel, including heterogeneity in both the long-run cointegrating vectors as well as heterogeneity in the dynamics associated with short-run deviations from these cointegrating vectors.

Response surfaces estimates for the Dickey-Fuller unit root test with structural breaks

The lack of suitable critical values for the Dickey-Fuller integrability test in finite-samples can drive researchers to spurious conclusions when using asymptotic critical values. In this paper we estimate response surfaces for the Dickey-Fuller unit root test with structural breaks that allow us to obtain approximate critical values for any sample size and break date.

Approximate method for the analysis of stability of beams under the action of a system of longitudinal and transverse forces

We propose an approximate method for the analysis of stability of uniplanar bending of beams under the action of a complex system of longitudinal and transverse forces, obtain formulas for the determination of the critical values of total longitudinal and transverse forces and the factor of stability of flange and rectangular beams that are convenient for practical applications, and present examples of the numerical analysis of actual beams. The critical values of the total forces obtained by the proposed method are compared with known results. The difference between the exact and approximate critical values does not exceed 2–7%. The experimental data enable us to recommend these approximate formulas for practical calculations of the stability of uniplanar bending of beams subjected to the combined action of complex systems of longitudinal and transverse forces.

Speed of invasion in lattice population models: pair-edge approximation

We propose a simple approach to approximating the speed of invasion in lattice population models. Approximate critical parameter values for successful invasion are then found by solving for zero wave speed. The approximation is based on describing the occupied region by the ordinary pair approximation, and using quasi-steady-state pair approximations to describe the leading edge of the wave front. We illustrate this idea using the basic contact process on the 1 and 2 dimensional lattice (with and without nearest-neighbor migration), finding very good agreement between the approximation and simulation results. The approximate critical values obtained by our approximation are significantly more accurate than those obtained by the ordinary pair approximation.

Power comparison of some tests for testing change point

This paper considers the problem of testing for a change point in variance of the sequence of normal random variables with unknown means. We construct three tests, namely, L-test derived from Lehmann's (1951) U-statistic, B-test obtained from Bayesian method and R-test derived by using the maximum likelihood ratio technique. Simulation studies were carried out to calculate the approximate critical values and the power of these tests. An empirical power comparison of these tests suggest that the power of the B-test is generally better than its competitors, the other two tests.

An Analysis-of-Means-Type Test for Variances from Normal Populations

After a brief review of the literature, a test for homogeneity of variances is presented. The test, which is the variances analog of the analysis of means (ANOM), can be presented in a graphical form. This graphical presentation makes it easy for practitioners to assess both practical and statistical significance. Tables of critical values are presented for balanced designs. A large-sample version of the test, which uses ANOM critical values, is also presented, as well as a version of the test for unbalanced designs for which approximate critical values can be calculated. Monte Carlo results indicate that this test has power comparable to well-known normal-based tests for homogeneity of variances.

An Analysis-of-Means-Type Test for Variances From Normal Populations

After a brief review of the literature, a test for homogeneity of variances is presented. The test, which is the variances analog of the analysis of means (ANOM), can be presented in a graphical form. This graphical presentation makes it easy for practitioners to assess both practical and statistical significance. Tables of critical values are presented for balanced designs. A large-sample version of the test, which uses ANOM critical values, is also presented, as well as a version of the test for unbalanced designs for which approximate critical values can be calculated. Monte Carlo results indicate that this test has power comparable to well-known normal-based tests for homogeneity of variances.

Nonparametric Tests for Comparing Umbrella Pattern Treatment Effects with a Control in a Randomized Block Design

This paper treats the problem of comparing umbrella pattern treatment effects with a control in a randomized block design. An extension of the Chen-Wolfe test (1993, Biometrics 49, 455-465) and its modification are proposed for both cases where the peak of the umbrella is known or unknown. Approximate critical values are given and the results of a Monte Carlo power study are presented.

Optimal testing for equicorrelated linear regression models

Recently, Knautz and Trenkler (1993) considered Christensen’s (1987) equicorrelated linear regression model as an example to show that S2 and\(\hat \beta \) are independent even though the disturbances are equicorrelated. This paper addresses the issue of testing for the equicorrelation coefficient in the linear regression model based on survey data. It computes exact and approximate critical values using Point optimal and F-test statistics, respectively. An empirical comparison of these critical values at five percent nominal level are presented to demonstrate the performance of the new tests.

Lag Order and Critical Values of the Augmented Dickey–Fuller Test

Response surface analysis is used to obtain approximate finite-sample critical values for the augmented Dickey–Fuller (ADF) test. Previous studies estimating the critical values for the test have generally ignored their possible dependence on the lag order. This study shows that the lag order, in addition to the sample size, can affect the finite-sample behavior of the test. The result points to the importance of correcting for the effect of lag order in applying the ADF test.

The application of the durbin-watson test to the dynamic regression model under normal and non-normal errors

Until recently, a difficulty with applying the Durbin-Watson (DW) test to the dynamic linear regression model has been the lack of appropriate critical values. Inder (1986) used a modified small-disturbance distribution (SDD) to find approximate critical values. King and Wu (1991) showed that the exact SDD of the DW statistic is equivalent to the distribution of the DW statistic from the regression with the lagged dependent variables replaced by their means. Unfortunately, these means are unknown although they could be estimated by the actual variable values. This provides a justification for using the exact critical values of the DW statistic from the regression with the lagged dependent variables treated as non-stochastic regressors. Extensive Monte Carlo experiments are reported in this paper. They show that this approach leads to reasonably accurate critical values, particularly when two lags of the dependent variable are present. Robustness to non-normality is also investigated.

Isolating One-Cell Interactions

Robust regression can be used to identify one-cell interactions in multiway factorial designs. One-cell interactions appear as mean slippage outliers to a low-order model, and robust regression gives these cells low mean weight. I use mean cell weight as a diagnostic criterion for one-cell interactions and provide approximate critical values for mean cell weights. Several examples illustrate the method.

Isolating One-Cell Interactions

Robust regression can be used to identify one-cell interactions in multiway factorial designs. One-cell interactions appear as mean slippage outliers to a low-order model, and robust regression gives these cells low mean weight. I use mean cell weight as a diagnostic criterion for one-cell interactions and provide approximate critical values for mean cell weights. Several examples illustrate the method.